```Question 288565
The division process is shown below in detail.

Original expression is:

{{{(z^3-z^2-4)/(z+i)}}}

First you divide {{{z}}} into {{{z^3}}} to get {{{z^2}}}.
Then you multiply {{{z+i}}} by {{{z^2}}} to get {{{z^3 + z^2i}}}.
Then you subtract {{{z^3 + z^2i}}} from {{{z^3 - z^2}}} to get {{{-z^2 - z^2i}}}.
Then you divide {{{z}}} into {{{-z^2}}} to get {{{-z}}}.
Then you multiply {{{z+i}}} by {{{-z}}} to get {{{-z^2 - zi}}}.
Then you subtract {{{-z^2 - zi}}} from {{{-z^2 - z^2i}}} to get {{{-z^2i + zi}}}.
Then you divide {{{z}}} into {{{-z^2i}}} to get {{{-zi}}}.
Then you multiply {{{z+i}}} by {{{-zi}}} to get {{{-z^2i - zi^2}}}.
Then you subtract {{{-z^2i - zi^2}}} from {{{-z^2i + zi}}} to get {{{zi + zi^2}}}.
Then you divide {{{z}}} into {{{zi}}} to get {{{i}}}.
Then you multiply {{{z+i}}} by {{{i}}} to get {{{zi + i^2}}}.
Then you subtract {{{zi + i^2}}} from {{{zi + zi^2}}} to get {{{zi^2 - i^2}}}.
Then you divide {{{z}}} into {{{zi^2}}} to get {{{i^2}}}.
Then you multiply {{{z+i}}} by {{{i^2}}} to get {{{zi^2 + i^3}}}.
Then you subtract {{{zi^2 + i^3}}} from {{{zi^2 - i^2}}} to get {{{-i^2 - i^3}}}.
Then you bring down the {{{-4}}} to get {{{-i^2 - i^3 - 4}}}.

Since {{{i^2}}} = {{{-1}}}, you can substitute in this last expression to get:

{{{-(-1) - (-1*i) - 4}}} which becomes:

{{{1 + i - 4}}} which becomes:

{{{i - 3}}}.

{{{(z^3-z^2-4)/(z+i)}}} = {{{z^2 - z - zi + i + i^2}}} with a remainder of {{{i - 3}}}.

To prove that this is correct, you need to multiply the answer by {{{z+i}}} and then add the remainder back in to see if you can duplicate the original expression.

You do that in the following manner.

{{{z^2 - z - zi + i + i^2}}} * {{{z+i}}} equals:

{{{z^2 - z - zi + i + i^2}}} * {{{z}}} plus:
{{{z^2 - z - zi + i + i^2}}} * {{{i}}}.

First we multiply {{{z^2 - z - zi + i + i^2}}} * {{{z}}}.

That becomes:

{{{z^3 - z^2 - z^2i + zi + zi^2}}}.

Then we multiply {{{z^2 - z - zi + i + i^2}}} * {{{i}}}.

That becomes:

{{{z^2i - zi - zi^2 + i^2 + i^3}}}.

{{{z^3 - z^2 - z^2i + zi + zi^2}}} and {{{z^2i - zi - zi^2 + i^2 + i^3}}} together to get:

{{{z^3 - z^2 - z^2i + zi + zi^2 + z^2i - zi - zi^2 + i^2 + i^3}}}.

Then we combine like terms to get:

{{{z^3 - z^2 + i^2 + i^3}}}.

{{{z^2i}}} and {{{-z^2i}}} canceled out.
{{{zi}}} and {{{-zi}}} canceled out.
{{{zi^2}}} and {{{-zi^2}}} canceled out.

You are left with:

{{{z^3 - z^2 + i^2 + i^3}}}.

Since {{{i^2}}} = {{{-1}}}, you can substitute in this expression to get:

{{{z^3 - z^2 + i^2 + i^3}}} = {{{z^3 - z^2 - 1 - i}}}.

You now need to add the remainder of {{{i - 3}}} back in.

You get:

{{{z^3 - z^2 - 1 - i}}} plus {{{i - 3}}} equals:

{{{z^3 - z^2 - 1 - i + i - 3}}}.

Combine like results to get:

{{{z^3 - z^2 - 4}}}.

The {{{-i}}} and the {{{i}}} canceled out.

Since this is the same as the original expression you started with, your division is confirmed as being successfully concluded.

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